Question

Prove the following identities 
$$cosecx(\sec x-1)-\cot x(1-\cos x)=\tan x-\sin x$$

Answer 1

$$\text{ LHS }=cosecx(\sec x-1)-\cot x(1-\cos x)$$
$$=\frac{1}{\sin x}(\frac{1}{\cos x}-1)-\frac{\cos x}{\sin x}(1-\cos x)$$
$$=\frac{1}{\sin x}\left(\frac{1-\cos x}{\cos x}\right)-\frac{\cos x}{\sin x}(1-\cos x)$$
$$=\left(\frac{1-\cos x}{\sin x}\right)(\frac{1}{\cos x}-\cos x)$$
$$=\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{1-{\cos }^{2}x}{\cos x}\right)$$
$$=\left(\frac{1-\cos x}{\sin x}\right)\left(\frac{{\sin }^{2}x}{\cos x}\right)$$
$$=(1-\cos x)\left(\frac{\sin x}{\cos x}\right)$$
$$=\frac{\sin x}{\cos x}-\sin x$$
$$=\tan x-\sin x$$
 = RHS
Hence proved.